# Factor Graph-Based Online Bayesian Identification and Component Evaluation for Multivariate Autoregressive Exogenous Input Models

## TL;DR

This paper introduces a new method for identifying parameters in multivariate autoregressive models using Bayesian message passing on factor graphs.

## Contribution

The novelty lies in the Forney-style factor graph representation and message-passing procedure for Bayesian parameter identification in these models.

## Key findings

- The proposed method demonstrates convergence on a simulated autoregressive system.
- It achieves strong predictive performance on a benchmark task.
- The approach reveals how parameter uncertainty affects predictive uncertainty.

## Abstract

We present a Forney-style factor graph representation for the class of multivariate autoregressive models with exogenous inputs, and we propose an online Bayesian parameter-identification procedure based on message passing within this graph. We derive message-update rules for (1) a custom factor node that represents the multivariate autoregressive likelihood function and (2) the matrix normal Wishart distribution over the parameters. The flow of messages reveals how parameter uncertainty propagates into predictive uncertainty over the system outputs and how individual factor nodes and edges contribute to the overall model evidence. We evaluate the message-passing-based procedure on (i) a simulated autoregressive system, demonstrating convergence, and (ii) on a benchmark task, demonstrating strong predictive performance.